Quadratic Equations

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A quadratic equation is defined as an equation of second degree where at least one variable or term is raised to a power of 2 and is written in the following Standard Form $a x^{2} + b x + c = 0$ , where and $c$ are all Real Numbers and $a \neq 0$ .

Examples of quadratic Equations are:

$x^{2} + 2 x - 24 = 0 3 a^{2} - 2 a + 2 = 0 5 p^{2} - p = 0 r^{2} = 200$

What is the standard form of a quadratic equation?

As mentioned above, the Standard Form of a quadratic equation is: $a x^{2} + b x + c = 0$

The values of $a, b$ and $c$ are all Real Numbers.

Any variable or constant multiplied by 0 equals 0. Therefore, the value of $a$ can never equal 0 ( $a \neq 0$ ), as this would transform the standard equation into a linear equation. See below:

$0 x^{2} + b x + c = 0 b x + c = 0$

How are quadratic equations solved?

Quadratic Equations are solved by determining the roots of the equation. These are its x-intercepts, where $y = 0$ e.g. the points at which the graph cuts through the x-axis. Finding these points can be done in one of the following ways:

Factoring

Factoring is when we determine which terms need to be multiplied to get a mathematical expression. Factoring quadratic equations can be done in the following ways:

Taking the greatest common factor (GCF)

Taking the greatest common factor is when we determine the highest term that evenly divides into all other terms.

$x^{2} + 4 x = 0$

Step 1: Find the greatest common factor by identifying the numbers and variables that each term has in common.

$x^{2} = x \cdot x 4 x = 4 \cdot x$

As we can see the most appearing variable is x, therefore making our GCF=x.

Step 2: Write out each term as a product of the greatest common factor and another factor, i.e. the two parts of the term. The other factor can be determined by dividing your term by your GCF.

$\frac{x^{2}}{x} = x ∴ x^{2} = x \cdot x \frac{4 x}{x} = 4 ∴ 4 x = x \cdot 4$

Step 3: Having rewritten your terms, rewrite your quadratic equation in the following form: $a b + a c = 0$

$x^{2} + 4 x = x (x) + x (4) = 0$

Step 4: Apply the law of distributive property and factor out your greatest common factor.

$x (x) + x (4) = x (x + 4)$

Step 5 (how to solve the quadratic equation): Equate the equation to 0 and solve for the

x-intercepts.

$x_{1} : x = 0 x_{1} = 0 x_{2} : x + 4 = 0 x_{2} = - 4$

Perfect square method

The perfect square method is when we transform a perfect square trinomial $a^{2} + 2 a b + b^{2}$ into a perfect square binomial ${(a + b)}^{2}$ or $a^{2} - 2 a b + b^{2}$ ,into ${(a - b)}^{2}$ .

$x^{2} + 14 x + 49 = 0$ Step 1: Transform your equation from standard form, $a x^{2} + b x + c = 0$ , into a perfect square trinomial, $a^{2} + 2 a b + b^{2}$ .

x^{2} + 14 x + 49 = x^{2} + 2 (x) (7) + 7^{2}

Step 2: Transform the perfect square trinomial into a perfect square binomial,

{(a + b)}^{2}

x^{2} + 2 (x) (7) + 7^{2} = {(x + 7)}^{2}

Step 3: Calculate the value of the x-intercept by equating the perfect square binomial to 0 and solving for x.

{(x + 7)}^{2} = 0 x + 7 = 0 x = - 7

Grouping

Grouping is when we group terms that have Common Factors before factoring. Let's look at how we do this. $2 x^{2} + 11 x + 12$ Step 1: List out the values of a, b and c.

$a = 2 b = 11 c = 12$

Step 2: Find the two numbers that product ac and also add to b.

$a c = 24 b = 11$

1 \times 24 = 24 2 \times 12 = 24 3 \times 8 = 24 3 + 8 = 11

The two numbers are therefore 3 and 8, as they can be used to add to 11. The other Factors of 24 can not be arranged in any way that would make them equal to 11.

Step 3: Use these Factors to separate the x-term (bx) in the original expression/equation. $2 x^{2} + 11 x + 12 = 2 x^{2} + 3 x + 8 x + 12$ Step 4: Use grouping to factor the expression.

(2 x^{2} + 3 x) + (8 x + 12) = x (2 x + 3) + 4 (2 x + 3) = (x + 4) (2 x + 3)

Step 5 (how to solve the quadratic equation): Equate the factored expression to 0 and solve for the x-intercepts.

$(x + 4) (2 x + 3) = 0 x_{1} : x + 4 = 0 x = - 4 x_{2} : 2 x + 3 = 0 x = - \frac{3}{2}$

Completing the square

Completing the Square is when we change the standard form of the quadratic equation into a perfect square with an additional constant. This means changing $a x^{2} + b x + c = 0$ into $a {(x + m)}^{2} + n$ , m is a real Number and n is a constant. They are calculated in the following way: $m = \frac{b}{2 a}$ and $n = c - (\frac{b^{2}}{4 a})$ .

$x^{2} + 6 x + 7$

Step 1 : List out the values of a, b and c.

a = 1 b = 6 c = 7

Step 2 : Calculate the value of m by using the following equation:

m = \frac{b}{2 a}

m = \frac{6}{2 (1)} = 3

Step 3: Calculate the value of n by using the following equation:

n = c - (\frac{b^{2}}{4 a})

n = 7 - (\frac{6^{2}}{4 (1)}) = 7 - 9 = - 2

Step 4: Substitute your calculated values and value of a into the following equation:

a {(x + m)}^{2} + n

1 {(x + 3)}^{2} - 2 = {(x + 3)}^{2} - 2

The rest of the steps below tell us how we'd solve for our x-intercepts. Step 5: Equate your equation to 0.

{(x + 3)}^{2} - 2 = 0 {(x + 3)}^{2} = 2 \sqrt{{(x + 3)}^{2}} = \pm \sqrt{2} x + 3 = \pm \sqrt{2} x_{1} = \sqrt{2} - 3 = - 1.59 x_{2} = - \sqrt{2} - 3 = - 4.41

Quadratic formula

The quadratic formula is a formula that uses the coefficients and constants of a quadratic equation to solve the equation by determining its x-intercepts/roots. It includes $\pm$ which indicates there are two solutions. The quadratic formula is: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

The discriminant of a quadratic formula is: $b^{2} - 4 a c$ which can tell us how many solutions the equation has. See below:

A positive discriminant means that the quadratic equation has two different real number solutions.
A negative discriminant means that none of the solutions are real numbers.
Real numbers are numbers that can be identified on a timeline. For example, infinity is not a real number because it doesn't have a measurable size and therefore can't be identified on a number line.
A discriminant that is equal to 0 means that the quadratic equation has a repeated real number solution.

We use the following symbol to represent the discriminant: $∆$

Quadratic Equations Parabola examples depending on discriminant StudySmarter A representation of how a parabola is drawn depending on the discriminant, Nicole Moyo-StudySmarter Originals

The following steps will show us how to solve a quadratic equation by using the quadratic formula:

$3 x^{2} - 4 x - 2 = 0$ Step 1: List out the values of a, b and c.

$a = 3 b = - 4 c = - 2$

Step 2: Calculate the value of the discriminant.

$∆ = {(- 4)}^{2} - 4 (3) (- 2) = 40$ Step 3: Substitute the values of a,b and c into the quadratic formula and solve for both roots/solutions.

$x_{1} = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} = \frac{- (- 4) + \sqrt{40}}{2 (3)} = \frac{2 + \sqrt{10}}{3} = 1.72 x_{2} = \frac{- b - \sqrt{b^{2} - 4 a c}}{2 a} = \frac{- (- 4) - \sqrt{40}}{2 (3)} = \frac{2 - \sqrt{10}}{3} = - 0.39$

Using a calculator

A calculator can be used to solve quadratic equations, but the method differs slightly according to the type of calculator that you use. The steps below show how to solve quadratic equations using a Casio Es Model calculator:

Step 1: Write out the values of a, b and c, from your quadratic equation.

Step 2: Select Mode and 5 to open the equation calculation mode.

Step 3: Select 3 to enter the quadratic equation solve mode.

Step 4: Plug in the values of a, b and c and press = to solve the quadratic equation.

What do quadratic equation graphs look like?

Quadratic equation Graphs are U-shaped curves, called parabolas. The equation of parabolas is commonly expressed as: $y = a x^{2} + b x + c$ . For example: $y = 3 x^{2} - 6 x + 5$

The parts of a quadratic equation graph are:

Axis of symmetry: $x = - \frac{b}{2 a}$
Vertex: calculated by substituting the axis of symmetry into the original equation.
Minimum or maximum point: The minimum point is the lowest point of the vertex and the maximum point is the highest point of the vertex.
Y Intercept: calculated by making x=0, in the original equation.
X Intercepts: Calculated by making y=0, in the original equation.

Quadratic Equations Labeled Parabola StudySmarter An illustration of a quadratic equation parabola, Nicole Moyo -StudySmarter Originals

Let us follow the steps below, to learn more about how a parabola is sketched:

$y = x^{2} - 4 x + 3$

Step 1: Solve for the axis of symmetry, by using this equation: $x = - \frac{b}{2 a}$

$x = \frac{- (- 4)}{2 (1)} = 2$

Step 2: Substitute the value of your axis of symmetry into your original equation to determine your vertex (the turning point of the graph).

$y = 2^{2} - 4 (2) + 3 = - 1$

Step 3: Find your y-intercept by making x=0.

$y = {(0)}^{2} - 4 (0) + 3 = 3$

Step 4: Find your x-intercepts, if any, by making y=0.

0 = x^{2} - 4 x + 3 = (x - 1) (x - 3) x_{1} = 1 x_{2} = 3

Step 5: List out and plot all your calculated points. Vertex: (2,-1) Y Intercept: (0,3) X Intercepts: (1,0) (3,0)

Quadratic Equations Parabola Graph Solution StudySmarter

Graph solution of parabola, Nicole Moyo-Study Smarter Originals

Quadratic Equations - Key takeaways

A quadratic equation is an equation of second degree.
The standard form of a quadratic equation is: $a x^{2} + b x + c = 0$ , where $a \neq 0$ .
Quadratic equations are solved by factorising, Completing the Square, using the quadratic formula and using a calculator.
Quadratic equation Graphs are called parabolas which are U-shaped curves.
The parts of a parabola are axis of symmetry, vertex, minimum or maximum point, y-intercept, and x-intercepts.

Frequently Asked Questions about Quadratic Equations

A quadratic equation is defined to be an equation of second degree where at least one variable or term is raised to a power of 2.

Examples of quadratic equations are: 0=2x^2 , -x^2+6x=-18, 4x^2=4x-1

Quadratic equations are solved in one of the following ways: factoring, completing the square and using the quadratic formula.

Flashcards in Quadratic Equations9 Start learning

What is factoring?

Factoring is when we determine which terms need to be multiplied together in order to get a mathematical expression.

How are quadratic equations solved?

Quadratic equations are solved by determining its roots

What are the roots of a quadratic equation?

The roots of a quadratic equation are its x-intercepts. They are the points where the graph cuts through the x-axis.

State three ways that quadratic equations can be factored.

Grouping
Completing the Square
Taking out the Greatest Common Factor

Briefly state what taking the greatest common factor is.

Taking the greatest common factor is when we determine the highest term that evenly divides into all other terms.

What is a quadratic equation?

A quadratic equation is an equation of 2nd degree.

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